Download Chapters 4-6: Estimation

Estimating µ after transforming data
For large sample sizes (n ≥ 30), we do not need to assume that the data {xi } are normal in order to find
a CI for µX . But when we have a small sample (n < 15) from a population which is clearly non-normal,
then a “transform to normality”, such as the Box-Cox family of transforms Y = X λ (in Chapter 4-6,
Normal Distribution notes). In many cases, one can not directly interpret the point estimate y or a CI
for µY .
1. Box-Cox transform Y = X λ with λ 6= 0
• The mean µX is always estimated by X̄, but you may want to use medians as a measure of
center due to skew or the presence of outliers.
1
1
2
1
2
λ
σX
(by the
• µX is estimated by y λ if xs 2 is “small”. This is because µYλ ≈ µX 1 + λ(λ−1)
2
2
µ
X
Delta Method).
1
• The median µ̃X is estimated by y λ if Y is normal. This is because medians transform to
medians.
• The back-transformed interval
1 sy λ
y − t1− α , n−1 √
, y + t1− α ,
2
2
n
sy
√
n−1
n
1 !
λ
1
– is a 100(1 − α2 ) CI for (µY ) λ .
– is a 100(1 − α2 ) CI for the median µ̃X if Y is normal.
– is an approximate 100(1 − α2 ) CI for µX when
s2
x2
is “small”
• When λ < 0, you’ll need to swap the endpoints of the CI
2. Box-Cox transform Y = ln(X) (λ = 0)
• The mean µX is always estimated by X̄, but you may want to use medians as a measure of
center due to skew or the presence of outliers.
2
σ2
• µX is estimated by exp(y) if xs 2 is “small”. This is because exp(µY ) ≈ µX exp − 2µX2
X
(by the Delta Method).
• The median µ̃X is estimated by exp(y) if Y is normal. This is because medians transform
1
to medians. By the rules of logs, exp(y) = (Î i xi ) n , which is the geometric mean of the
untransformed data.
• The back-transformed interval
exp y − t1− α ,
2
sy
√
n−1
n
, exp y + t1− α ,
2
sy
√
n−1
n
.
– is a 100(1 − α2 ) CI for exp (µY ), the population geometric mean of X.
– is a 100(1 − α2 ) CI for the median µ̃X if Y is normal.
– is an approximate 100(1 − α2 ) CI for µX when
s2
x2
is “small”